Po1757, po1762 looks like something to do with transmission

It looks like you have a problem with the torque converter clutch solenoid or valve.
If torque converter stay applies while you come to a stop. it's like a standard transmission that you start in fourth gear.
If you remove the transmission pan, look for pieces of metal or dirt, debris.
I suggest you change the transmission torque converter solenoid and retest.
you will need to refill the transmission with the right transmission fluid.

All they told you was that "something" was missing? Did they tell you it was the drain plug bolt?

Hopefully it is just something in the shift linkage and not inside the transmission.
Set the parking brake and block the wheels. Check the shift linkage on the transmission and see if if something has blocked it or if it is bent. If there is nothing obvious, disconnect the linkage and see if it is stuck on the shifter side or the transmission side.
If on the shifter side, you may be able to track down the problem. If the problem is in the transmission, you may have to have a mechanic or transmission specialist take a look at it.

could be clogged transmission filter or shift valves in valve body sticking go to auto parts look for transmission additives to help free shift valves and buy transmission fluid for high mileage transmissions the fluid will recondition swell up transmission seal help transmission shift better.

this noise is most likely coming from the transmission pump, its a sign that something is on its way out have transmission shop look at it,cheaper to replace with used one

The pan has 14 bolt holes,you will have to remove,the transmission mount to jack the trans up to access the bolts.

You will need to look under the vehicle at the transmission and verify if there is transmission case damage,a broken transmission line ect. if your son ran over something and you can see the damage then adding fluid will cause it to just leak back out of the case. You will probably need it to be towed to the mechanic if that is the case. If you try and drive it in that condition you may cause permanent damage to the vehicle's tranny. And transmission fluid is usually added in the same tube that contains the dip stick to check the level of the fluid. Also power steering fluid will also look like transmission fluid and could be a power steering leak.

CODES PO505 IDLE AIR CONTROL MOTOR CIRCUIT. P2781 AND PO1757 DID NOT SEE THEM IN CODE MANUAL.CODE PO505 WHAT ITS TELLING YOU.THAT YOUR STEPPER MOTOR COIL CIRCUIT IS SHORTED TO SYSTEM POWER B+ OR STEPPER MOTOR COIL NUMBER 1 2 3 4 CIRCUIT OPEN OR SHORTED TO GROUND.

I would recommend you take it to 3 different repair shops to get quotes on the cost of repairs.
Sounds internal to me .
Automatic transmissions are best left to the Pro's to fix , You need alot of tools and a very very very clean workshop for starters.
Hope this helps , Sorry I couldn't offer a quick fix.

Where is this wire? is it at the transmission range selector switch on top of the trans where the shift cable connects?